Percolation pp 25-43 | Cite as

Some Basic Techniques

  • Geoffrey Grimmett


The probability θ(p), that the origin is in an infinite open cluster, is a non-decreasing function of p. This is intuitively obvious since an increase in the value of p leads generally to an increase in the number of open edges of L d , thereby increasing the number and length of open paths from the origin. Another way of putting this is to say that {|C| = ∞} is an increasing event, in the sense that if ω ∈ {|C| = ∞} then ω′ ∈ {|C| = ∞} whenever ωω′. With such an example in mind we make the following definition. The event A in F is called increasing if I A (ω) ≤ I A (ω′) whenever ωω′, where I A is the indicator function of A. We call A decreasing if its complement A c is increasing.


Basic Technique Disjoint Path Reliability Theory Open Path Open Edge 
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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Geoffrey Grimmett
    • 1
  1. 1.School of MathematicsUniversity of BristolBristolEngland

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